Overlaying 2 transparent images on another texture
You're mixing up shaders with texture compositing operations.

The shaders are the ones with the green sockets, and you should never connect other colours to the green sockets or vice-versa.

The yellow and gray sockets are used for colours and values respectively. In practice it means you'll use the yellow sockets to add colour textures to your shaders inputs, and grayscale textures (or single values) to control properties.

(that's a gross over-simplification, but it will help to have a minimum grasp of what you should/shoudn't connect to each type of socket).

With the above in mind, remember that you can mix and combine textures BEFORE connecting them to a shader. So in this case you should use mixRGB nodes to combine them and plug the result to the shader.

In your example of brick texture and grunge textures it would be just matter of blending the first grunge texture over the brick texture (using a mix node if the grunge one has alpha channel will do, but you could also use a different blending and connect the second texture's alpha to the Fac input to drive the mix).

Once you did that, the output of that mix will be textures 1 and 2 blended, so you only need to add another mix and blend the third texture over.

That will be your "combined" texture that you will plug to the colour socket of your shader (most likely a diffuse shader). 

You can add further work to your shader if you need some glossiness or other surface properties to your material, but that's basically the procedure of combining textures.

*) Note that mixing two shaders (one transparent and an emissive/reflective one) to create a material with transparency based on a texture's alpha is a completely different procedure.